Calculator
Example data table
Sample values for f(x)=sin(x) (useful for data mode with a small stencil).
| x | f(x) |
|---|---|
| 0.0 | 0.0000000000 |
| 0.5 | 0.4794255399 |
| 1.0 | 0.8414709848 |
| 1.5 | 0.9974949866 |
| 2.0 | 0.9092974268 |
Formula used
This tool approximates the m-th derivative at x0 using a weighted sum of function values on a stencil:
f(m)(x0) \approx \sum\limits_{i=0}^{n-1} wi\, f(xi)
The stencil points are x_i = x0 + k_i h, where k_i are integer offsets. The weights w_i are computed with a numerically stable recursive method that works for both uniform and non-uniform stencils.
- Truncation error: for smooth functions, increasing stencil size increases the formal order in h.
- Round-off error: extremely small h can amplify floating-point noise; choose h sensibly.
How to use this calculator
- Choose Analytic function or Tabulated data mode.
- Set the evaluation point x0 and the step size h.
- Select derivative order m and a target accuracy order.
- Pick a scheme: central (balanced), forward, or backward.
- Keep Auto stencil unless you need a custom pattern.
- Click Compute to view the result above the form.
- Use the CSV/PDF buttons to export the stencil and output.
Purpose and scope of high-order schemes
High-order finite differences approximate derivatives using many nearby samples, raising accuracy without symbolic calculus. In physics workflows, these derivatives feed flux laws, curvature terms, and stability checks. This calculator covers arbitrary derivative order and flexible stencils for smooth fields.
Derivative order versus accuracy order
Choose the derivative order m (first, second, third, and beyond) and the target accuracy order p. For smooth data, truncation error typically scales like O(h^p). A 5-point central first-derivative stencil is 4th-order accurate, while 7-point can reach 6th order.
Stencil design, point count, and boundaries
Central schemes balance points around x0 and usually minimize bias in the interior. Near boundaries, forward or backward stencils avoid missing values. A practical rule is to use enough points to support your requested m and p; higher orders typically need wider stencils.
Weight computation and numerical stability
Weights w_i are generated from the stencil nodes x_i using a stable recursion (Fornberg-style). It supports uniform and non-uniform spacing, so stretched meshes and irregular sensors can be handled. On a uniform grid, weights reproduce polynomials up to the stencil degree.
Truncation error versus unresolved structure
Raising p reduces bias only if the function is smooth across the full stencil span. If the signal contains sharp features, high-order stencils may oscillate or overshoot. Compare a moderate stencil against a higher-order stencil at the same h to see whether the estimate stabilizes. If not, reduce stencil width or use a one-sided option.
Step size selection and round-off limits
Choosing h is a balance: too large increases truncation error, too small amplifies cancellation and floating-point noise. A practical starting range is 10^{-3} to 10^{-2} times the local length scale, then halve h and watch for a plateau. For higher m, the optimum often shifts toward larger h.
Working with tabulated measurements
In data mode, supply ordered (x,f) pairs that cover the stencil window around x0. Keep units consistent and avoid duplicated x values. If measurements are noisy, prefer moderate target orders and consider smoothing upstream, documenting any filter bandwidth or window size.
Documentation and exportable results
The CSV and PDF outputs preserve inputs, stencil offsets, coordinates, sampled values, and weights, making results reproducible. When reporting a derivative, record m, the stencil pattern, h, and the chosen order. Include the exported table as supplemental material when needed. This makes comparisons across solver versions and measurement campaigns far easier later.
FAQs
1) What does “high-order” mean in this tool?
It means the leading truncation error decays quickly with h, often like O(h^p). Achieving larger p generally requires more stencil points and careful step-size selection.
2) Should I always choose the maximum target order?
Not always. Very high order can magnify measurement noise and round-off, especially for larger m. Start moderate, then increase order only if results remain stable across small changes in h and stencil size.
3) Why do central stencils often perform better?
Symmetry around x0 cancels certain error terms and reduces bias for smooth interior data. Near boundaries, you may need forward or backward stencils because symmetric points are unavailable.
4) How can I pick a good step size h quickly?
Compute with several h values (e.g., halve h each time). Look for a region where the derivative estimate changes slowly. If values become erratic, increase h or lower the target order.
5) Can I differentiate irregularly spaced data?
Yes. In tabulated mode, weights are computed from the actual x_i values. Avoid tightly clustered points mixed with wide gaps, because conditioning can worsen and weights may become extreme.
6) What if the function is not smooth in the stencil window?
High-order finite differences assume smoothness. If a discontinuity or corner lies inside the stencil, the numerical derivative may be meaningless. Move x0, reduce stencil width, or use methods designed for shocks or jumps.
7) What exactly is included in the exports?
Exports include your inputs, stencil coordinates, function values, computed weights, and the final derivative estimate, plus simple diagnostics. This is enough to reproduce the result and document your numerical choices in reports.